Real-rooted atlas

A086645 → basis A013609 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(6 - 6\,x - 7\,n + 13\,n x + 2\,n^{2} - 9\,n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(24 + 48\,x + 8\,x^{2} - 28\,n - 32\,n x + 24\,n x^{2} + 8\,n^{2} - 12\,n^{2} x - 38\,n^{2} x^{2} + 8\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(6\,x + 14\,x^{2} + 4\,x^{3} - 4\,n x - 4\,n x^{2} + 8\,n x^{3} - 4\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2246
38321252
43219221632472
5128102419201024804176
651251201344012800400019236064
7204824576829441146886720013824448305728
881921146884730888601607526403010564390410242554752

All 15 rows

Technical details
Node id
n-7fb60f4a5335b07a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-0f919292b8973744

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      2,       4;
   3:      8,      32,       12;
   4:     32,     192,      216,       32;
   5:    128,    1024,     1920,     1024,       80;
   6:    512,    5120,    13440,    12800,     4000,      192;
   7:   2048,   24576,    82944,   114688,    67200,    13824,      448;
   8:   8192,  114688,   473088,   860160,   752640,   301056,    43904,    1024;
   9:  32768,  524288,  2555904,  5767168,  6758400,  4128768,  1204224,  131072,   2304;
  10: 131072, 2359296, 13271040, 35782656, 52715520, 43794432, 19869696, 4423680, 373248, 5120;
%F Formula
The row polynomials satisfy (2*n^2-7*n+6+(2*n^3-9*n^2+13*n-6)*x)*P(n,x) = (8*n^2-28*n+24+(8*n^3-12*n^2-32*n+48)*x+(12*n^3-38*n^2+24*n+8)*x^2)*P(n-1,x) - ((4*n-6)*x+(4*n^2+4*n-14)*x^2+(8*n^2-8*n-4)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 28*n + 24 + (8*n^3 - 12*n^2 - 32*n + 48)*x + (12*n^3 - 38*n^2 + 24*n + 8)*x^2)*P[n - 1, x] - ((4*n - 6)*x + (4*n^2 + 4*n - 14)*x^2 + (8*n^2 - 8*n - 4)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6 + (2*n^3 - 9*n^2 + 13*n - 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 28*n + 24 + (8*n^3 - 12*n^2 - 32*n + 48)*x + (12*n^3 - 38*n^2 + 24*n + 8)*x^2)*v[i - 1] - ((4*n - 6)*x + (4*n^2 + 4*n - 14)*x^2 + (8*n^2 - 8*n - 4)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6 + (2*n^3 - 9*n^2 + 13*n - 6)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023